Before you start: Most of what you learn here is unimportant and is definitely harder then what will be in the next post. Don't worry if your head hurts when learning this part, we will return to these topics later, and also we'll learn shortcuts for a lot of the math. Why learn this stuff first, then? To explain that would require you already knowing calculus, so I can't yet, sorry :P. Just know that it'll get easier for a bit.
Recap on What A Function Is (skip this if you already know)
A 'function' is basically an equation. A generic function is represented by f(x) (but it doesn't have to be 'f', it could be g(x), h(x), potato(x), etc...). Functions aren't that complicated, it's just another way of writing something like y=5x+2; you would write f(x)=5x+2. Why is this better than just using y? Because we can say something like this: f(1) = 5*1+2 and f(2) = 5*2+2. You can even get more complicated, and do things like f(x+h) = 5*(x+h)+2.IMPORTANT: 'f' is not a variable, DO NOT try to use the distributive property (f(x+h) is NOT f(x)+f(h)) If this confuses you, I recommend writing it as f[x] or f{x} at first to remind yourself.
What A Limit Is
If you don't know what a function is, read the previous section, it's important.Imagine a function
Mathematically, we write it like this:
We would solve this limit by realizing that we can factor out the 5 to get
There will be three main topics you have to worry about for Calculus AB, usually taught in this order: Limits>Derivatives>Integrals. However, because none of you are in Algebra II, and because the following was the order I learned it, I will actually be teaching it in this order: Derivatives>Limits>Integrals. Right now, you do not need to know how to calculate a limit yet. Just know that one way to solve a limit is to try to get rid of whatever is making the denominator equal 0.
What A Tangent Line Is, Why It Matters, And How To Find Its Slope
Tangent Line: A line that touches a curve at exactly one point. Here is an example:
The red line is tangent to the blue curve.
Why do we care about a tangent line? Imagine a car's position is defined by the function f(t)=5t+2 where t is seconds. When t=0, the car is at position 2, when t=1, the car is at position 7, and when t=x the car is at position 5x+2. What if we wanted to know how fast the car was? Well, the speed turns out to be the slope - 5! But what if the car traveled with a position defined by the function g(t)=5t^2+2? How fast is it moving now? The equation is no longer a line, it's a curve (a parabola), so it doesn't have a slope. However it does have tangents, and tangents are lines, which have slopes. The speed of a car at any time t is actually the slope of the tangent line at t.
Interlude: This may be confusing, that's fine, don't worry about it - just ask me to explain it better by finding me at support or contacting me through one of the methods I gave you - I can do a better job explaining this in-person than I can through text. We're almost done with this post, we've just got to get through this last part.
How do we find it's slope? Well, there's another type of line called a 'secant' line - I only told you its name just so you're not confused if you ever read about a secant line later, it's really quite unimportant. A 'secant line' is a line that intersects a line at two points. Let's call these points 'x' and 'x+h' (because the second point is 'h' away from the first point 'x'). The definition of a slope is:
Of course, we don't care about a secant line. Why did we learn this, then? Well, what we're trying to find is the slope of a tangent line, which is defined as the line that intersects the curve at only one point. The secant line intersects the curve at two points, which we called 'x' and 'x+h'. What if h was 0? Then both points would be 'x', which is only one point - we have the tangent line! So the equation of the tangent line is
His graph shows the green line (secant) having the distance between 'x' and 'x+h' (or in the graph's case P and Q) move infinitely close together until it's virtually identical to the tangent line.
Back to the problem of finding our car's speed. We defined its position as g(t)=5t^2+2. Oh no! We're using g(t) instead of f(x)! I guess we can't solve it :( ... Just kidding. It doesn't matter what you call the variables, we could have just as easily defined the slope of the tangent line to be
The limit of 10t+5h is easy to solve - just plug in '0' for 'h' to get 10t. What does '10t' mean? It's the equation for the slope of a tangent line. Looking at a graph of a parabola:
Epilogue
Congratulations, you've made it! You successfully survived the first post. In my opinion, this post is definitely one of the hardest pasts of calculus - not because the topic is hard, but because you're introduced to so many concepts at one, and since you've not been in Algebra II yet some of the math skills we use may seem foreign. Don't worry, it'll get better - I can look at the equationProblems (optional but recommended)
What is the equation for the slope of the tangent line of:
5x+7
2x^2+3x+9
x^3
(x+1)^2
If you solved all of those, try to solve:
Hint: Don't try to solve this if you haven't figured out the 'trick' I mentioned, you'll have a bad time. I should also probably mention that
Graphs from (in order of appearance):
coolmath.com
Paul's Online Calculus Notes
webformulas.com
